As for the Waif, it wouldn't surprise me if this is some kind of test for her too, but it looks like Arya is out. She couldn't bring herself to kill an innocent person, so what place does she have with the Faceless Men now? I doubt this is exactly how it will go in the books; she needs to take away a little something more from them for it to be worth anything, but the show is trying to cut corners to reach an ending at this point.
The summary for next week:
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The last two bits are obvious enough. What about the first two? Who would the High Sparrow's new target be? The Hound, maybe? I tend to doubt that because I doubt the Hound and the High Sparrow will personally interact in this episode, if the Hound is on the Quiet Isle or elsewhere Brienne might encounter him. Who else? Pycelle? Olenna? We see Olenna chastising Cersei in the preview for putting a pox on both their houses. And who is Jaime's hero? There aren't many heroes running around between King's Landing and Riverrun. Is it just the Blackfish? We saw them speaking in the preview. What makes the Blackfish such a hero? Just the fact that he rallied the troops to hold Riverrun? Maybe a bit of hyperbole.